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What is the polynomial of the smallest degree with integer coefficients and with zeros \[3,{\text{ }}1 + i, - 1\]?
$\left( {\text{1}} \right){\text{ x}}_{}^{\text{4}}{\text{ + 4x}}_{}^{\text{3}}{\text{ + 3x}}_{}^{\text{2}}{\text{ + 2x - 6}}$
$\left( {\text{2}} \right){\text{ x}}_{}^{\text{4}}{\text{ - 4x}}_{}^{\text{3}}{\text{ + 2x}}_{}^{\text{2}}{\text{ + 3x - 6}}$
$\left( {\text{3}} \right){\text{ x}}_{}^{\text{4}}{\text{ - 4x}}_{}^{\text{2}}{\text{ + 5x - 6}}$
$\left( {\text{4}} \right){\text{ x}}_{}^{\text{4}}{\text{ - 4x}}_{}^{\text{3}}{\text{ + 3x}}_{}^{\text{2}}{\text{ + 2x - 6}}$

seo-qna
Last updated date: 25th Jun 2024
Total views: 405.3k
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Answer
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Hint: We know that no real roots exist in roots always. So one more root always remains present that is, $1 - i$.

Formula used: Formula p(x) = (x - a)(x - b)(x - c)

Complete step-by-step answer:
It is given that the question, total number of the roots $ = 4$
Thus we can say that the minimum degree of the polynomial would be $ = 4$
And the required polynomial can be found out by applying the formula $p(x) = (x - a)(x - b)(x - c)$
Here \[a{\text{ }} = - 1\],\[b = 3\], and $c = 1 - i$
Therefore, the required polynomial by doing multiplication would be-
$(x + 1)(x - 3)({x^2} - 2x + 2)$
On multiplying the first terms we get,
$ = (x_{}^2 - 3x + x - 3)(x_{}^2 - 2x + 2)$
Let us subtract the middle terms we get,
$ = (x_{}^2 - 2x - 3)(x_{}^2 - 2x + 2)$
On multiply the term we get,
$ = (x_{}^4 - 2x_{}^3 + 2x_{}^2 - 2x_{}^3 + 4x_{}^2 - 4x - 3x_{}^2 + 6x - 6)$
On simplifying the terms in the same coefficients we get,
$ = (x_{}^4 - 4x_{}^3 + 3x_{}^2 + 2x - 6)$
Hence the required polynomial is $(x_{}^4 - 4x_{}^3 + 3x_{}^2 + 2x - 6)$

Thus the $\left( {\text{4}} \right)$ option is correct.

Note: It is to be kept in mind that we need to find out the sum of the roots of the equation and then by applying the formula of polynomial function we can solve the question easily. These questions are easy but most of the students make mistakes while doing calculation of the required polynomial. So keep focus while doing it and try to avoid such mistakes.
A polynomial can be considered as expressions that consist of two or more algebraic terms. Polynomials contain constants, variables and exponents.
Polynomials can be classified into three types such as monomial, binomial and trinomial.
Monomial is an expression which contains only one term. For example \[5x\]
Binomial is an expression which contains two terms. Example $ - 5x + 3$
Trinomial can be regarded as an expression which contains three terms. For example $4x_{}^2 + 9x + 7$
There are four main polynomial operations, that is, addition of polynomials, subtraction of polynomials, multiplication of polynomials and division of polynomials.